| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s + 2·5-s − 4·6-s + 6·7-s + 4·8-s + 9-s + 4·10-s − 10·11-s − 4·12-s + 14·13-s + 12·14-s − 4·15-s + 8·16-s − 6·17-s + 2·18-s − 6·19-s + 4·20-s − 12·21-s − 20·22-s − 2·23-s − 8·24-s + 2·25-s + 28·26-s + 2·27-s + 12·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s − 1.63·6-s + 2.26·7-s + 1.41·8-s + 1/3·9-s + 1.26·10-s − 3.01·11-s − 1.15·12-s + 3.88·13-s + 3.20·14-s − 1.03·15-s + 2·16-s − 1.45·17-s + 0.471·18-s − 1.37·19-s + 0.894·20-s − 2.61·21-s − 4.26·22-s − 0.417·23-s − 1.63·24-s + 2/5·25-s + 5.49·26-s + 0.384·27-s + 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.848073487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.848073487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) | |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 8 T^{3} + 31 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ac_c_ai_bf |
| 7 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} + 71 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ag_s_abk_ct |
| 11 | $D_4\times C_2$ | \( 1 + 10 T + 74 T^{2} + 32 p T^{3} + 125 p T^{4} + 32 p^{2} T^{5} + 74 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.k_cw_no_cax |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 198 T^{3} + 1004 T^{4} + 198 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.g_bt_hq_bmq |
| 19 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} - 120 T^{3} - 721 T^{4} - 120 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.g_s_aeq_abbt |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 40 T^{2} - 4 T^{3} + 1315 T^{4} - 4 p T^{5} - 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.c_abo_ae_byp |
| 29 | $C_2^3$ | \( 1 + 49 T^{2} + 1560 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_bx_a_cia |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.aq_ey_abmu_kfy |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 384 T^{3} + 2036 T^{4} + 384 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.e_cb_ou_dai |
| 41 | $D_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 910 T^{3} + 7144 T^{4} + 910 p T^{5} + 125 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.k_ev_bja_kou |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 100 T^{2} + 528 T^{3} + 6411 T^{4} + 528 p T^{5} + 100 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.g_dw_ui_jmp |
| 47 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1168 T^{3} + 9982 T^{4} - 1168 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.aq_ey_absy_oty |
| 53 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ack_a_jtb |
| 59 | $D_4\times C_2$ | \( 1 + 32 T + 452 T^{2} + 3908 T^{3} + 28807 T^{4} + 3908 p T^{5} + 452 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.bg_rk_fui_bqpz |
| 61 | $D_4\times C_2$ | \( 1 - 24 T + 337 T^{2} - 3480 T^{3} + 29016 T^{4} - 3480 p T^{5} + 337 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ay_mz_afdw_bqya |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 90 T^{2} + 1212 T^{3} - 18625 T^{4} + 1212 p T^{5} + 90 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.as_dm_buq_abboj |
| 71 | $C_2^3$ | \( 1 + 10 T + 50 T^{2} - 920 T^{3} - 9641 T^{4} - 920 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.k_by_abjk_aogv |
| 73 | $D_4\times C_2$ | \( 1 - 14 T + 98 T^{2} - 840 T^{3} + 7031 T^{4} - 840 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ao_du_abgi_kkl |
| 79 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 2310 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_ado_a_dkw |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 44 T^{3} - 8594 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.e_i_abs_amso |
| 89 | $D_4\times C_2$ | \( 1 + 26 T + 194 T^{2} - 1204 T^{3} - 29153 T^{4} - 1204 p T^{5} + 194 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ba_hm_abui_abrdh |
| 97 | $D_4\times C_2$ | \( 1 - 30 T + 234 T^{2} + 2436 T^{3} - 54481 T^{4} + 2436 p T^{5} + 234 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.abe_ja_dps_adcpl |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.334675012977016515514965554410, −8.062830878127096718825810681685, −8.029890256533448531844223727629, −8.008531530578253259793031593650, −7.58853288618617541058436681877, −6.85735893281206722992952062687, −6.75383827789170679945376645481, −6.61045044593689406200844282986, −6.40520609989305718624686698562, −5.80677338089602130973340743443, −5.64040193476888752394374980832, −5.61890095834128384723405248002, −5.48796365204231895919436538379, −4.86893579834841808612123608843, −4.80480854814687732993644536374, −4.47620856720446478946972429399, −4.44826226547852969822794463832, −3.85340499560761168925782122543, −3.70459306793396372443369287639, −3.05832744956174382926638588942, −2.68156981808866275252987492001, −2.24682492671853828606758675339, −1.84043054836887611524201195272, −1.53751776265104037776807544680, −0.907048390563699537393718210796,
0.907048390563699537393718210796, 1.53751776265104037776807544680, 1.84043054836887611524201195272, 2.24682492671853828606758675339, 2.68156981808866275252987492001, 3.05832744956174382926638588942, 3.70459306793396372443369287639, 3.85340499560761168925782122543, 4.44826226547852969822794463832, 4.47620856720446478946972429399, 4.80480854814687732993644536374, 4.86893579834841808612123608843, 5.48796365204231895919436538379, 5.61890095834128384723405248002, 5.64040193476888752394374980832, 5.80677338089602130973340743443, 6.40520609989305718624686698562, 6.61045044593689406200844282986, 6.75383827789170679945376645481, 6.85735893281206722992952062687, 7.58853288618617541058436681877, 8.008531530578253259793031593650, 8.029890256533448531844223727629, 8.062830878127096718825810681685, 8.334675012977016515514965554410
